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G = C23.7C24  order 128 = 27

7th non-split extension by C23 of C24 acting via C24/C22=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C23.7C24, C24.471C23, 2+ (1+4).13C22, C4○C2≀C22, (C2×D4)⋊25D4, (C2×Q8)⋊20D4, (C22×C4)⋊6D4, C2≀C22⋊6C2, C4.88C22≀C2, C23⋊C4⋊7C22, C23.26(C2×D4), C2.C25⋊3C2, (C23×C4)⋊26C22, (C2×D4).41C23, C4○(C23.7D4), C23.7D4⋊6C2, C22⋊C4.1C23, C22≀C2⋊24C22, C22.19C24⋊5C2, C22.41(C22×D4), C42⋊C2⋊12C22, (C22×C4).616C23, C23.C23⋊18C2, C22.D4⋊29C22, (C2×C4).27(C2×D4), C2.62(C2×C22≀C2), (C2×C4○D4).111C22, SmallGroup(128,1757)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C23.7C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — C2.C25 — C23.7C24
Lower central
C1 — C2 — C23 — C23.7C24
Upper central
C1 — C4 — C22×C4 — C23.7C24
Jennings
C1 — C2 — C23 — C23.7C24

Subgroups: 772 in 383 conjugacy classes, 106 normal (10 characteristic)
C1, C2, C2 [×11], C4, C4 [×3], C4 [×13], C22 [×3], C22 [×27], C2×C4 [×12], C2×C4 [×37], D4 [×42], Q8 [×10], C23, C23 [×6], C23 [×9], C42 [×3], C22⋊C4 [×6], C22⋊C4 [×12], C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C22×C4 [×10], C2×D4 [×9], C2×D4 [×24], C2×Q8 [×3], C2×Q8 [×6], C4○D4 [×40], C24, C23⋊C4 [×12], C42⋊C2 [×3], C4×D4 [×6], C22≀C2 [×6], C4⋊D4 [×3], C22⋊Q8 [×3], C22.D4 [×6], C23×C4, C2×C4○D4 [×3], C2×C4○D4 [×6], 2+ (1+4) [×4], 2+ (1+4) [×3], 2- (1+4) [×3], C23.C23 [×3], C2≀C22 [×4], C23.7D4 [×4], C22.19C24 [×3], C2.C25, C23.7C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C23.7C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=c, ab=ba, faf=ac=ca, ede=ad=da, ae=ea, ag=ga, ebe=bc=cb, fdf=bd=db, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Permutation representations
►On 16 points - transitive group 16T223
Generators in S16
(1 9)(2 10)(3 11)(4 12)(5 16)(6 13)(7 14)(8 15)

(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)

(1 3)(2 4)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)

(1 16)(2 13)(3 14)(4 15)(5 9)(6 10)(7 11)(8 12)

(1 16)(2 13)(3 14)(4 15)(5 11)(6 12)(7 9)(8 10)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
 
G:=sub<Sym(16)| (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12), (1,16)(2,13)(3,14)(4,15)(5,9)(6,10)(7,11)(8,12), (1,16)(2,13)(3,14)(4,15)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
 
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12), (1,16)(2,13)(3,14)(4,15)(5,9)(6,10)(7,11)(8,12), (1,16)(2,13)(3,14)(4,15)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
 
G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,16),(6,13),(7,14),(8,15)], [(1,11),(2,12),(3,9),(4,10),(5,16),(6,13),(7,14),(8,15)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,3),(2,4),(5,14),(6,15),(7,16),(8,13),(9,11),(10,12)], [(1,16),(2,13),(3,14),(4,15),(5,9),(6,10),(7,11),(8,12)], [(1,16),(2,13),(3,14),(4,15),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])
 
G:=TransitiveGroup(16,223);
 

►On 16 points - transitive group 16T270
Generators in S16
(5 7)(6 8)(13 15)(14 16)

(9 11)(10 12)(13 15)(14 16)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)

(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)

(1 7)(2 8)(3 5)(4 6)(9 13)(10 14)(11 15)(12 16)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
 
G:=sub<Sym(16)| (5,7)(6,8)(13,15)(14,16), (9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,7)(2,8)(3,5)(4,6)(9,13)(10,14)(11,15)(12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
 
G:=Group( (5,7)(6,8)(13,15)(14,16), (9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,7)(2,8)(3,5)(4,6)(9,13)(10,14)(11,15)(12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
 
G=PermutationGroup([(5,7),(6,8),(13,15),(14,16)], [(9,11),(10,12),(13,15),(14,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)], [(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)], [(1,7),(2,8),(3,5),(4,6),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])
 
G:=TransitiveGroup(16,270);
 

Matrix representation ►G ⊆ GL4(𝔽5) generated by

4200
0100
1401
1410
,
1300
0400
0101
0110
,
4000
0400
0040
0004
,
1000
0100
0001
0010
,
4030
4044
0010
1410
,
4030
0041
0010
0110
,
3000
0300
0030
0003
G:=sub<GL(4,GF(5))| [4,0,1,1,2,1,4,4,0,0,0,1,0,0,1,0],[1,0,0,0,3,4,1,1,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[4,4,0,1,0,0,0,4,3,4,1,1,0,4,0,0],[4,0,0,0,0,0,0,1,3,4,1,1,0,1,0,0],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3] >;
 

32 conjugacy classes

class 1 2A2B2C2D2E···2L4A4B4C4D4E4F···4M4N···4S
order122222···2444444···44···4
size112224···4112224···48···8

32 irreducible representations

dim1111112224
type+++++++++
imageC1C2C2C2C2C2D4D4D4C23.7C24
kernelC23.7C24C23.C23C2≀C22C23.7D4C22.19C24C2.C25C22×C4C2×D4C2×Q8C1
# reps1344316334

In GAP, Magma, Sage, TeX 

C_2^3._7C_2^4
 
% in TeX
 
G:=Group("C2^3.7C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1757);
 
// by ID
 
G=gap.SmallGroup(128,1757);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,248,718,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=c,a*b=b*a,f*a*f=a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,a*g=g*a,e*b*e=b*c=c*b,f*d*f=b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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